Problem 17 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then which one of the following statements is a tautology? \(\quad\) (a) \((\mathrm{p} \vee \mathrm{r}) \rightarrow(\mathrm{p} \wedge \mathrm{r})\) (b) \((\mathrm{p} \wedge \mathrm{r}) \rightarrow(\mathrm{p} \vee \mathrm{r})\) (c) \(\mathrm{p} \wedge \mathrm{r}\) (d) \(\mathrm{p} \vee \mathrm{r}\)

Step-by-Step Solution

Answer

Statement (b) is a tautology.

Step 1: Interpret the given condition

We know given that \( q \) is false and the statement \( p \wedge q \leftrightarrow r \) is true. For \( p \wedge q \) to be true, \( p \) needs to be true, but \( q \) is false. Therefore, \( p \wedge q \) is false. Since \( p \wedge q \leftrightarrow r \) is true, it means \( r \) must also be false because \( \ot(p \wedge q) \) means \( r \) must be false to keep the biconditional true.

Step 2: Evaluate statement (a)

Consider statement \(( p \vee r) \rightarrow(p \wedge r)\). If both \( p \) and \( r \) are false, then \( p \vee r \) is false. Thus, \( (p \vee r) \rightarrow(p \wedge r) \) is true by the implication rule \( \lnot A \rightarrow B \).

Step 3: Evaluate statement (b)

Consider statement \((p \wedge r) \rightarrow(p \vee r)\). Since \( p \wedge r \) is false, the implication is true because a false antecedent implies any consequent.

Step 4: Evaluate statement (c)

Consider statement \(p \wedge r\). Since both \( p \) and \( r \) are false, \( p \wedge r \) is false.

Step 5: Evaluate statement (d)

Consider statement \(p \vee r\). Since both \( p \) and \( r \) are false, \( p \vee r \) is false.

Key Concepts

Logical Implications

Logical implications are statements that express a relationship between two propositions. They are often written in the form \( A \rightarrow B \), where \( A \) is called the antecedent and \( B \) is the consequent. A logical implication is considered true in every case except when \( A \) is true and \( B \) is false. This means if the antecedent is true but the consequent is not, the whole statement fails. However, if the antecedent is false, the implication is automatically true, regardless of the truth value of the consequent. This seemingly counterintuitive rule is fundamental in the field of logic and underlies many logical arguments and deductions.

• If \( A \) is true and \( B \) is true, \( A \rightarrow B \) is true.
• If \( A \) is true and \( B \) is false, \( A \rightarrow B \) is false.
• If \( A \) is false, \( A \rightarrow B \) is always true.

Understanding this framework is essential for evaluating expressions involving logical implications, as was done in the exercise solution.

Biconditional Statement

A biconditional statement is a logical statement that involves a combination of two conditional statements. It is denoted by \( A \leftrightarrow B \) and reads as "\( A \) if and only if \( B \)." For the entire biconditional to be true, both \( A \) and \( B \) must share the same truth value. In other words, either both components need to be true, or both need to be false.
This is a stricter form of implication that only holds under these conditions.

• \( A \) and \( B \) both true: \( A \leftrightarrow B \) is true.
• \( A \) and \( B \) both false: \( A \leftrightarrow B \) is true.
• Mixed truth values (one true, one false): \( A \leftrightarrow B \) is false.

In the exercise, the condition \( p \wedge q \leftrightarrow r \) was true, so both sides needed to have the same truth value in the end.

Truth Values in Logic

Truth values in logic refer to the assignment of either 'true' (typically represented as 1) or 'false' (represented as 0) to a proposition. Each logical statement or proposition can be evaluated to one of these truth values. Logical operations such as AND (\( \wedge \)), OR (\( \vee \)), and NOT (\( \lnot \)) manipulate these truth values to evaluate complex statements.

Truth values are critical in determining the validity of logical arguments and understanding how complex expressions derive their final truth assignments. For instance:

• \( p \) being true and \( q \) being false implies \( p \wedge q \) is false, since both need to be true for \( \wedge \) to be true.
• If \( r \) is false, then any expression where \( r \) is alone with an \( \wedge \) operation will also be false.

By examining truth values, one can solve logical puzzles and check for the truthfulness of statements within the framework of logic.

Logical Conjunction and Disjunction

Logical conjunction and disjunction are fundamental operations in logic used to combine simple statements into more complex ones. Conjunction is represented by the \( \wedge \) symbol and is akin to the logical "AND" operation. For any conjunction \( A \wedge B \), the statement is true only if both \( A \) and \( B \) are true.

Disjunction, on the other hand, is represented by the \( \vee \) symbol and acts as the logical "OR". For a disjunction \( A \vee B \), the statement is true if at least one of the statements, \( A \) or \( B \), is true.

• Conjunction \( A \wedge B \): true only if both \( A \) and \( B \) are true.
• Disjunction \( A \vee B \): true if either \( A \), \( B \), or both are true.

Understanding these operations helps in analyzing and constructing logical statements as seen in the exercise, where the truth values of conjunctions and disjunctions were key in evaluating each option.